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December 2010 Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data
Yehua Li, Tailen Hsing
Ann. Statist. 38(6): 3321-3351 (December 2010). DOI: 10.1214/10-AOS813

Abstract

We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies.

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Yehua Li. Tailen Hsing. "Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data." Ann. Statist. 38 (6) 3321 - 3351, December 2010. https://doi.org/10.1214/10-AOS813

Information

Published: December 2010
First available in Project Euclid: 20 September 2010

zbMATH: 1204.62067
MathSciNet: MR2766854
Digital Object Identifier: 10.1214/10-AOS813

Subjects:
Primary: 62J05
Secondary: 62G20 , 62M20

Keywords: Almost sure convergence , Functional data analysis , ‎kernel‎ , local polynomial , nonparametric inference , principal components

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 6 • December 2010
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